(a) Let f(x, y) = P(x, y)i +Q(x, y)j be a vector field, where P and Q are continuous differentiable functions. Let C be a smooth curve parameterized by x= x(t), y = y(t) where ast≤ b with position vector r(t) = x(t)i +y(t)j. Show that ff.dr=-ff.dr -C с Hence prove that ff.dr=0 C (b) Use divergence theorem to calculate the surface integral fF.dS; that is, calculate the flux S of F across S. F(x, y, z) = (y² cos z + xy²)i +(x³e²) j+[sin(3y) + 1x²z]k, S is the surface of the solid bounded by the paraboloid z = x² + y² and the plane z = 6
(a) Let f(x, y) = P(x, y)i +Q(x, y)j be a vector field, where P and Q are continuous differentiable functions. Let C be a smooth curve parameterized by x= x(t), y = y(t) where ast≤ b with position vector r(t) = x(t)i +y(t)j. Show that ff.dr=-ff.dr -C с Hence prove that ff.dr=0 C (b) Use divergence theorem to calculate the surface integral fF.dS; that is, calculate the flux S of F across S. F(x, y, z) = (y² cos z + xy²)i +(x³e²) j+[sin(3y) + 1x²z]k, S is the surface of the solid bounded by the paraboloid z = x² + y² and the plane z = 6
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(a) Let f(x, y) = P(x, y)i +Q(x, y)j be a vector field, where P and Q are continuous
differentiable functions. Let C be a smooth curve parameterized by x= x(t), y = y(t)
where ast≤b with position vector r(t) = x(t)i + y(t)j.
Show that ff.dr=-ff.dr
C
Hence prove that ff.dr = 0
(b) Use divergence theorem to calculate the surface integral fF.dS; that is, calculate the flux
S
of F across S. F(x, y, z) = (y² cos z + xy²)i +(x³e¯²)j +[sin(3y) + 1x²z]k, S is the surface
of the solid bounded by the paraboloid z = x² + y² and the plane z = 6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F425ee8a1-7c2f-4657-9e12-faf7ddcef3f4%2Fc65c5338-3e16-41d5-b65d-7efcfc751f59%2Fjigernl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let f(x, y) = P(x, y)i +Q(x, y)j be a vector field, where P and Q are continuous
differentiable functions. Let C be a smooth curve parameterized by x= x(t), y = y(t)
where ast≤b with position vector r(t) = x(t)i + y(t)j.
Show that ff.dr=-ff.dr
C
Hence prove that ff.dr = 0
(b) Use divergence theorem to calculate the surface integral fF.dS; that is, calculate the flux
S
of F across S. F(x, y, z) = (y² cos z + xy²)i +(x³e¯²)j +[sin(3y) + 1x²z]k, S is the surface
of the solid bounded by the paraboloid z = x² + y² and the plane z = 6
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